TABLE OF CONTENTS In Chapter 7, we learned about the stability and error properties of numerical schemes and we have provided a methodology for the quantitative estimation of the associated errors. We have also derived some guidelines on mesh sizes to achieve a preset level of accuracy, particularly with the very demanding simulation of time-dependent problems.
In addition, we studied a certain number of second order schemes for the convection equations, in particular the leapfrog and the celebrated Lax-Wendroff schemes, both based on central difference formulas, and we have briefly mentioned the second order upwind scheme of Warming and Beam.
At this stage you might have asked yourself about the eventual existence of other schemes? This would be totally justified, as we have noticed already in Chapter 4, that an unlimited number of finite difference formulas can be defined and that for every mathematical model, an unlimited number of schemes could indeed be written down. However, they will not be equally acceptable in practice, as we have learned from Chapter 7. Stability limits, error properties can vary significantly between various schemes and it would be very useful to rely on guidelines for the evaluation of the best-adapted schemes for a given application.
In response to this objective, we will introduce in this chapter a general approach to derive conditions on families of schemes having a predetermined support and order of accuracy. The presented methodology will allow you to either select a new scheme with preset properties, or to evaluate an...
